Díaz Díaz, Jesús Ildefonso and Lions, J.L. (1999) On the approximate controllability for some explosive parabolic problems. In Optimal control of partial differential equations. International Series of Numerical Mathematics (133). Birkhäuser Verlag AG, Basel, pp. 115-132. ISBN 3-7643-6151-4
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Abstract
We consider in this paper distributed systems governed by parabolic evolution equations which can blow up in finite time and which are controlled by initial conditions. We study here the following question : Can one choose the initial condition in such a way that the solution does not blow up before a given time T and which is, at time T, as close as we wish from a given state ? Some general results along these lines are presented here for semilinear second order parabolic equations as well as for a non local nonlinear problem. We also give some results proving that "the more the system will blow up" the "cheaper" it will be the control.
| Item Type: | Book Section |
|---|---|
| Additional Information: | International Conference on Optimal Control of Partial Differential Equations.CHEMNITZ, GERMANY . APR 20-25, 1998 |
| Subjects: | Sciences > Mathematics > Differential geometry |
| ID Code: | 15662 |
| References: | Bardos, C. and Tartar, L., 1973, Sur l'unicité rétrograde des équations paraboliques et quelques questions voisines, Arch. Rat'ion. Mech. Analysis, 50, pp. 10-25 Blayo, E., Blu m, J. and Verron, J., 1998, Assimilation variationelle de données en Océanographie et réduction de la dimension de l'espace de contróle.In Équations aux dérivées partielles et applicatons:Articles dédiés à Jacques-Louis Lions, GauthierVillars, Paris, pp. 199-220. Cazenave, Th. and Haraux, A., 1990, Introduction aux problemes d'évolution semilinéaires, Ellipses, Paris. Díaz, J.r., 1991, Sur la contrólabilité approchée de inéquations variationelles et d'autres problemes paraboliques non linéaire, C. R. Acad. Scie. de Paris, 1312, Série I, pp. 519-522. Díaz, J.r., Henry, J. and Ramos, A.M., 1998, On the Approximate Conrollability of Sorne Semilinear Parabolic Boundary-Value Problems, Appl. Math. Optim.,37,pp.71-97. Díaz, J.I. and Lions, J.-L., 1998, Sur la contrólabilité de problemes paraboliques avec phénomimes d'explosion, C. R. Acad. Seie. de PaTis. To appear. Fabre, C., Puel, J.P., and Zuazua, E., 1995, On the density of the range of the semigroup for semilinear heat equations. In Control and Optimal Design of DistTibuted Parameter Systems. Springer-Verlag, New York, IMA Volumes #70, pp. 73-92. Henry, J., 1978, Etude de la controlabilité approchée de certaines équations paraboliques, These d'Etat, Université de Paris VI. Lattes, R. and Lions, J.L., 1967, Méthode de quasiréversibilité et applications, Dunod, Paris. Le Dimet, F.X. and Charpentier, 1., 1998, Méthodes du second ordre en assimilation de données. In Équations aux dérivées partielles et applicatons: Articles dédiés a Jacques-Louis Lions, Gauthier-Villars, Paris, pp. 623-640. Lions, J.L. 1968, Controle optimal de systemes gov.vernés par des équations à derivées partielles, Dunod, Paris. Lions, J.L. 1983, Controle des systemes distribués singuliers, Gauthier-Villars, Bordas, Paris. Lions, J.L. and Zuazua, E., 1997, The cost of controlling unstable systems: time irreversible systems, Revista Matemática de la Univ. Complutense de Madrid,10,pp. 481-523. Vrabie, U., 1995, Compactness Methods for Nonlinear Evolutions, Pitman Monographs, Longman, Harlow. |
| Deposited On: | 18 Jun 2012 10:23 |
| Last Modified: | 18 Jun 2012 10:23 |
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